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For 0 ≤ α ≤ 1 , let L α (x) = ∑ n ≤ x λ (n) n α , where λ (n) is the Liouville function. Then famous criteria of Pólya and Turán claim that the eventual sign constancy of each of L 0 (x) and L 1 (x) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of L α (x) , L α ′ (x) , L α ″ (x) for all 1 / 2 < α < 1 is equivalent to the Riemann hypothesis. Analogous equivalent conditions for the Riemann hypothesis are formulated in terms of partial sums of the Möbius function as well. Our criteria indicate a new tendency concerning the values of the Liouville and Möbius functions in stark contrast with their random behavior. Further generalizations are given in two essential directions, namely that equivalent criteria are developed for any quasi-Riemann hypothesis, and we allow the partial sums to be supported on semigroups after sifting out integers divisible by members of a set of prime numbers whose size is under control. Our negativity conditions for the quasi-Riemann hypothesis exhibit a curious rigidity in two respects, first it doesn't matter which particular primes we use to sift out, but only the size of the set, second the sign of the partial sums is unaltered with respect to two successive derivatives. Finally, other competing necessary or sufficient conditions for a quasi-Riemann hypothesis are studied regarding the monotonicity, zeros and sign changes of partial sums of Dirichlet series. [ABSTRACT FROM AUTHOR]

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at s = 1. [ABSTRACT FROM AUTHOR]

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis [ABSTRACT FROM AUTHOR]

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series. [ABSTRACT FROM AUTHOR]